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Notes on a Family of Inhomogeneous Linear Ordinary Differential Equations

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NOTES ON A FAMILY OF INHOMOGENEOUS LINEAR ORDINARY DIFFERENTIAL EQUATIONS

FENG QI

Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China; College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China;

Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China

JING-LIN WANG

Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China

BAI-NI GUO

School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

Abstract. In the note, the authors find several simple and explicit forms for a family of inhomogeneous linear ordinary differential equations studied in “D. Lim,Differential equations for Daehee polynomials and their applications, J. Nonlinear Sci. Appl. 10 (2017), no. 4, 1303–1315; Available online at

http://dx.doi.org/10.22436/jnsa.010.04.02”.

1. Motivation and main results

In [3, Theorem 1], it was obtained inductively and recurrently that the differential equations

∂nF(t, x) ∂ tn =

1 tn

" n X i=0

ai(n, x) +

bi(n, x) ln(1 +t)

t

1 +t i#

F(t, x), n≥0 (1.1)

E-mail addresses:qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com, jing-lin.wang@hotmail.com, bai.ni.guo@gmail.com, bai.ni.guo@hotmail.com. 2010Mathematics Subject Classification. 11B73, 11M35, 26A48, 26D07, 34A05, 44A10. Key words and phrases. simple form; explicit form; differential equation; Lerch transcendent; logarithmic function.

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have the same solution

F(t, x) =ln(1 +t) t (1 +t)

x, (1.2)

wherea0(n, x) = (−1)nn! and b0(n, x) = 0 for alln≥0,

aj(n, x) = (x)j(−1)n−j(n−j)! n−j X ij−1=0

n−j−ij−1

X ij−2=0

· · ·

n−j−ij−1−···−i2

X ii=0

(n−ij−1− · · · −i1−j+ 1)

and

bj(n, x) = j−1 X k=0

j−1 Y i=0

(x−i)(−1)n−j(nj)! x−k

n−j X ij−1=0

n−j−ii−1

X ij−2=0

· · ·

n−j−ij−1−···−i2

X i1=0

(n−ij−1− · · · −i1−j+ 1)

for 1≤j≤n+ 1, and (x)n =Q n−1

k=0(x−k) is the falling factorial.

It is clear that the above expressions foraj(n, x) and bj(n, x) are very difficult to compute by hand or by computer. The derivation of the quantitiesaj(n, x) and bj(n, x) in [3] is much long and tedious.

The aim of this paper is to alternatively supply several new, simple, and explicit expressions for aj(n, x) and bj(n, x). In other words, the aim of this paper is to alternatively provide several new, simple, and explicit forms for the family of differential equations in (1.1).

Our main results can be stated as the following theorems.

Theorem 1. Forn≥0, the function F(t, x)defined in (1.2)satisfies ∂nF

∂ tn = (−1) nn!

tn ( n

X i=0

(−1)i(x)i i!

"

1− 1

ln(1 +t) n−i X k=1 1 k

t

1 +t

k# t

1 +t i)

F (1.3)

and

∂nF ∂ tn =

n! (1 +t)n

( n X i=0

(−1)i (x)n−i (n−i)!

"

1− 1

ln(1 +t) i X k=1 1 k

t

1 +t

k#1 +t t

i) F.

Theorem 2. Let

Φ(z, s, a) =

X k=0

zk

(a+k)s, a6= 0,−1, . . .

denote the Lerch transcendent. Then

∂nF(t, x) ∂ tn =

(−1)nn! (1 +t)n−x+1

n X i=0

(−1)i(x)i i! Φ

t

1 +t,1, n−i+ 1

and

∂nF(t, x) ∂ tn =

n! (1 +t)n−x+1

n X

i=0

(−1)i (x)n−i (n−i)!Φ

t

1 +t,1, i+ 1

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2. A lemma

In order to prove our main results in Theorems 1 and 2, we need the following lemma.

Lemma 1([7, Theorem 2]). Let

f(x) =

  

lnx

x−1, 0< x6= 1; 1, x= 1.

Forn≥0, the nth derivative off(x)can be computed by

f(n)(x) =

      

(−1)nn! (x−1)n+1

"

lnx− n X k=1 1 k

x1 x

k#

, 0< x6= 1;

(−1)n n!

n+ 1, x= 1

(2.1)

and

f(n)(x) =

    

(−1)n n! xn+1Φ

x−1

x ,1, n+ 1

, 0< x6= 1;

(−1)n n!

n+ 1, x= 1.

(2.2)

3. Proofs of Theorems 1 and 2

We now start out to prove our main results in Theorems 1 and 2.

Proof of Theorem 1. By (2.1), it is straightforward that

∂nF(t, x) ∂ tn =

n X i=0

n i

ln(1 +t) t

(n−i)

[(1 +t)x](i)

= n X i=0 n i

(−1)n−i(n−i)! tn−i+1

"

ln(1 +t)− n−i X k=1 1 k t 1 +t

k#

(x)i(1 +t)x−i

= (1 +t)

xln(1 +t)

t

(−1)nn! tn

n X

i=0

(−1)i(x) i i!

"

1− 1

ln(1 +t) n−i X k=1 1 k

t

1 +t

k# t

1 +t i

= (−1)nn! tn

( n X i=0

(−1)i(x)i i!

"

1− 1

ln(1 +t) n−i X k=1 1 k

t

1 +t

k# t

1 +t i)

F(t, x)

and

∂nF(t, x) ∂ tn =

n X i=0 n i

ln(1 +t) t

(i)

[(1 +t)x](n−i)

= n X i=0 n i

(1)ii! ti+1

"

ln(1 +t)− i X k=1 1 k t

1 +t k#

(x)n−i(1 +t)x−n+i

= ln(1 +t) t(1 +t)n−x

n X i=0

n i

(1)ii! ti

"

1− 1

ln(1 +t) i X k=1 1 k t

1 +t k#

(x)n−i(1 +t)i

= F(t, x) (1 +t)n

n X i=0

(−1)i n! (n−i)!

"

1− 1

ln(1 +t) i X k=1 1 k t

1 +t k#

(x)n−i 1 +t

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= n! (1 +t)n

( n X

i=0

(−1)i (x)n−i (n−i)!

"

1− 1

ln(1 +t) i X k=1 1 k

t

1 +t

k#1 +t t

i) F(t, x).

The proof of Theorem 1 is complete.

Proof of Theorem 2. By (2.2), it is immediate that

∂nF(t, x) ∂ tn =

n X i=0

n i

ln(1 +t) t

(n−i)

[(1 +t)x](i)

= n X i=0

n i

(−1)n−i (n−i)! (1 +t)n−i+1Φ

t

1 +t,1, n−i+ 1

(x)i(1 +t)x−i

= (−1) nn!

(1 +t)n−x+1 n X i=0

(−1)i(x)i i! Φ

t

1 +t,1, n−i+ 1

and

∂nF(t, x) ∂ tn =

n X i=0

n

i

ln(1 +t) t

(i)

[(1 +t)x](n−i)

= n X i=0

n i

(−1)i i! (1 +t)i+1Φ

t

1 +t,1, i+ 1

(x)n−i(1 +t)x−n+i

= n!

(1 +t)n−x+1 n X i=0

(−1)i (x)n−i (n−i)!Φ

t

1 +t,1, i+ 1

.

The proof of Theorem 2 is complete.

4. Remarks

Finally we give several remarks about our main results mentioned and verified above.

Remark 1. Comparing (1.1) with (1.3) reveals that the formulas

ai(n, x) = (−1)n−i n!

i!(x)i and

bi(n, x) =−ai(n, x) n−i X k=1 1 k

t

1 +t k

= (−1)n−i+1n! i!(x)i

n−i X k=1 1 k

t

1 +t k

should be valid for alli, n≥0.

Remark 2. The idea of this paper comes from the articles [1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and the closely related references therein.

References

[1] B.-N. Guo and F. Qi,Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math.272(2014), 251–257; Available online athttp://dx.doi.org/10.1016/j.cam.2014.05.018.

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[3] D. Lim,Differential equations for Daehee polynomials and their applications, J. Nonlinear Sci. Appl.10(2017), no. 4, 1303–1315; Available online at http://dx.doi.org/10.22436/ jnsa.010.04.02.

[4] F. Qi,Explicit formulas for the convolved Fibonacci numbers, ResearchGate Working Paper (2016), available online athttp://dx.doi.org/10.13140/RG.2.2.36768.17927.

[5] F. Qi and B.-N. Guo, Explicit formulas and nonlinear ODEs of generating functions for Eulerian polynomials, ResearchGate Working Paper (2016), available online athttp://dx. doi.org/10.13140/RG.2.2.23503.69288.

[6] F. Qi and B.-N. Guo,Identities of the Chebyshev polynomials, the inverse of a triangular ma-trix, and identities of the Catalan numbers, Preprints2017, 2017030209, 21 pages; Available online athttp://dx.doi.org/10.20944/preprints201703.0209.v1.

[7] F. Qi and B.-N. Guo, Some properties of a solution to a family of inhomogeneous linear ordinary differential equations, Preprints2016, 2016110146, 11 pages; Available online at

http://dx.doi.org/10.20944/preprints201611.0146.v1.

[8] F. Qi and B.-N. Guo, Some properties of the Hermite polynomials and their squares and generating functions, Preprints2016, 2016110145, 14 pages; Available online athttp://dx. doi.org/10.20944/preprints201611.0145.v1.

[9] F. Qi and B.-N. Guo, Viewing some nonlinear ODEs and their solutions from the angle of derivative polynomials, ResearchGate Working Paper (2016), available online at http: //dx.doi.org/10.13140/RG.2.1.4593.1285.

[10] F. Qi and B.-N. Guo, Viewing some ordinary differential equations from the angle of de-rivative polynomials, Preprints 2016, 2016100043, 12 pages; Available online at http: //dx.doi.org/10.20944/preprints201610.0043.v1.

[11] F. Qi, J.-L. Wang, and B.-N. Guo,Simplifying two families of nonlinear ordinary differential equations, Preprints2017, 2017030192, 6 pages; Available online athttp://dx.doi.org/10. 20944/preprints201703.0192.v1.

[12] F. Qi and J.-L. Zhao,Some properties of the Bernoulli numbers of the second kind and their generating function, J. Differ. Equ. Appl. (2017), in press. ResearchGate Working Paper (2017), available online athttp://dx.doi.org/10.13140/RG.2.2.13058.27848.

[13] F. Qi and J.-L. Zhao, The Bell polynomials and a sequence of polynomials applied to differential equations, Preprints 2016, 2016110147, 8 pages; Available online at http: //dx.doi.org/10.20944/preprints201611.0147.v1.

[14] J.-L. Zhao, J.-L. Wang, and F. Qi,Derivative polynomials of a function related to the Apostol– Euler and Frobenius–Euler numbers, J. Nonlinear Sci. Appl.10(2017), no. 4, 1345–1349; Available online athttp://dx.doi.org/10.22436/jnsa.010.04.06.

c

2017 by the authors; licensee Preprints, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license http://creativecommons.org/licenses/by/4.0/).

URL:https://qifeng618.wordpress.com

URL:http://orcid.org/0000-0001-6725-533X

References

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